The equivariant index factorizes into contributions coming from each fixed point. Each of these contributions is an infinite series which depends on a choice of regularization. To patch all terms, and recover the correct equivariant index of \((E^\bullet ,\eth )\) , one needs to choose the regularization consistently. A procedure for certain cases where the complex can be lifted over a Sasakian \(S^1\) -fibration of \(X\)  has been given in [1]}. However, it is limited to specific ASD/SD distributions in four dimensions and not every compact, simply-connected four-manifold can be lifted to a Sasakian \(S^1\) -fibration (e.g. \(S^4\) ).
